If the model for \(Y_i\)  in terms of \(\lbrace X_i,Z_i\rbrace \)  is linear, and the BLUP consistently estimates the conditional mean function, \(E(X_i\mid X_i^*, Z_i)\) , then the resultant estimator \(\widehat{\Theta }_\text{RC}\)  will be consistent, provided mild regularity conditions. If the model for \(Y_i\)  is non-linear, or the mean function is not consistently estimated, we cannot generally guarantee consistency. In certain non-linear models, however, specific claims can be made. In a log-linear model, with the correct mean function, all slope parameters will be consistently estimated. In a logistic regression model bias will generally be reduced and, if interest is in the estimated probabilities rather than the parameters, the approximation is often quite accurate [1]}. Thus the regression calibration estimators are regarded as approximately consistent [1]}. Under regularity conditions, after certain transformations, they are asymptotically normal [3]}.
